The Republican speaker of the House of Representatives, John Boehner, is pursued by the press. Photograph: J Scott Applewhite/AP

The fate of a deal to resolve America's fiscal cliff crisis hung in the balance on Tuesday, in spite of an overwhelming vote in the Senate hours earlier in favour of a compromise bill aimed at ending the long-running saga.

President Barack Obama hailed the Senate vote and called on the House of Representatives to act "without delay". But senior figures in the House, where the Republicans have a majority, expressed dissatisfaction with the deal and it was unclear on Tuesday what its fate would be.

Obama was looking for a deal to be in place to calm Wall Street before it re-opened on Wednesday. But the office of the Republican House speaker, John Boehner, could not confirm whether he would even schedule a vote on the bill. Boehner, in theory, should be able to push the bill through with a combination of moderate Republicans and Democrats, even in the face of a revolt by Tea Party-backed members of Congress.

Eric Cantor, the Republican leader in the House, said he did not support the bill, further raising questions about how Congress will address automatic spending cuts and tax increases that began to take effect just after midnight.

Both Republicans and Democrats expressed unhappiness with the bill. Democrats accused Obama of giving too much ground to the Republicans, setting too high a threshold for tax increases to kick in. Republicans claimed the deal failed to address America's burgeoning deficit.

One of the conservative House Republicans, Michele Bachmann, indicated she would vote against, saying the bill increased revenue without dealing with spending. "It is a cynical, planned move by the president," she told Congress.

Republicans scheduled at least two closed-door meetings on Tuesday to discuss the issues with their leadership. The White House dispatched vice-president Joe Biden to Congress to persuade liberal Democrats in the House to back the bill.

Democrats in the House emerged from the briefing with Biden pledging support, and urging Republican colleagues to accept the bill unamended.

Minority House leader Nancy Pelosi said the legislation sent from the Senate represented a "historic" bi-partisan compromise. She also piled pressure on Boehner to allow the measures to go to a vote in the House, noting that he had previously suggested that any bill from the Senate would be put in front of representatives.

"That is what he said, that is what we expect. That is what the American people deserve," Pelosi said.

The bill passed by the Senate, with 89 senators in favour and eight against, is a messy, short-term deal that raises taxes on the wealthiest but postpones for two months any consideration of spending cuts. The vote came at 2am on Tuesday, too late to prevent the country breaching the midnight fiscal cliff deadline. Taxes for all Americans automatically went up on 1 January and all federal programmes, from defence to welfare, faced deep spending cuts.

To reverse this, the House must also vote for the bill, which confines tax rises to individuals earning $400,000 or more a year and households earning $450,000 or more. It postpones spending cuts for two months, to allow further negotiations. Estate tax also rises, to 40% from 35%, but inheritances below $5m are exempted from the increase. Benefits for the unemployed are extended for another year.

The House presents a much bigger hurdle than the Senate, not only because the Republicans have a majority but because of the presence of a bloc of Tea Party-backed Republicans. The Republicans have 241 members to the Democrats' 191.

Boehner's instinct is towards compromise but he has had trouble keeping the Tea Party bloc under control. Theoretically, he has enough votes to push a bill through, but he is coming up for re-election as Speaker and will not want to alienate conservative Republicans.

The deal was thrashed out in the past few days between Biden and the Republican leader in the Senate, Mitch McConnell. It partly fulfills one of Obama's election campaign promises: to raise taxes on the wealthiest. But Obama was forced to compromise: he wanted higher taxes to kick in at $250,000 a year.

In a statement, the White House welcomed the compromise: "For the first time in 20 years, Congress will have acted on a bipartisan basis to vote for significant new revenue. This means millionaires and billionaires will pay their fair share to reduce the deficit through a combination of permanent tax rate increases and reduced tax benefits."

The statement added that 98% of families would not face tax rises, assuming the House also voted for the deal.

Obama had wanted, as part of a deal, an assurance from the Republicans that they would not mount another showdown in a few months' time, when he has to raise the debt ceiling. But he failed to get this, ensuring yet another confronation between the White House and Repubicans in Congress in late February and early March.

The Republican senator John McCain, who voted for the deal, said: "Despite my disappointment with many aspects of this agreement, I cannot in good conscience stand by and watch taxes go up on all Americans.

"However, it should be embarrassing to all of us that it took until the last hours of the last day of the year to address an issue we should have dealt with months ago. This marks another sad chapter in what has been the least productive Congress since 1947."

Details of the deal

• Taxes will rise for individuals earning more than $400,000 and households that make more then $450,000, with the rate rising to a maximum of 39.6% from the current 35%. Capital gains and dividends in excess of those amounts would be taxed at 20%, up from 15%

• Estate tax will rise to 40% on the portion of estates exceeding $5m in value. At the insistence of Republicans, the $5m threshold would rise each year with inflation.

• Tax breaks will be maintained for families with children, for low-earning taxpayers and for those with a child in college. The alternative minimum tax, which was due to expand to affect an estimated 28 million households for the first time in 2013, with an average increase of more than $3,000, remains at its present level.

• The deal leaves untouched a scheduled 2 percentage point increase in the payroll tax, ending a temporary reduction enacted two years ago to help revive the economy.

• Benefits for the long-term unemployed, which were about to expire for an estimated two million jobless Americans, will be extended for a year.

• A 27% cut in fees for doctors who treat Medicare patients will be postponed for a year. Also included in the deal is a provision to prevent a threatened spike in milk prices.